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Astronomical Applications Department, U.S. Naval Observatory - lyap2 (Page 10)

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Astronomical Applications Department, U.S. Naval Observatory - lyap2
orbits. Each of these period n islands has asso-
ciated with it a similar hierarchy of higher-order
island chains, and so on.
Consider an outermost intact invariant sur-
face. A sufficient increase in the magnitude of
the perturbations will produce a tear in this sur-
face. Under dynamical evolution of the sur-
face, this tear will propagate and reproduce
over the surface, producing a fractal distribu-
tion (Cantor set) of holes or gaps, known as a
cantorus (Percival, 1979; MacKay et al., 1984).
The surface is not completely destroyed, but
neither is it any longer impervious to diffusion
of orbits across it. There is a countable infinity
of holes. The stronger the perturbation(s), the
more "porous" is the cantorus. Thus, it is pos-
sible for an orbit to be trapped in a region of
phase space, enclosed by a cantorus of small
porosity, for some time before encountering a
"hole" and escaping into a more unstable
region. Some orbits may escape quickly; oth-
ers will be trapped for some time. Given an ini-
tial population of such orbits, one would expect
a distribution of escape times ­ perhaps Gaus-
sian. The same argument can be used for orbits
near the secondary islands, which themselves
will be enclosed by cantori (and higher-order
islands, etc.).
Analytic calculation of diffusion rates
across hierarchies of cantori is very difficult.
As a trajectory passes through a cantorus, it
may be derailed to "shadow" a higher-order
island chain, which itself has a grid of cantori
gaps, and so on. Numerically, it can be shown
that the distribution of orbits initially in the
neighborhood of an island remaining in that
neighborhood (i.e., the "survival" probability) is
i t
for long times t (normalized by the orbit
period), where
1.4 for a small, isolated
island (Chirikov and Shepelyansky, 1984b;
Karney, 1983; Lichtenberg and Lieberman,
1992). For the main island of the standard
map (Chirikov, 1979),
1.45 (Chirikov and
Shepelyansky, 1984a; Murray, 1991). It
would be reasonable to wonder if the tori sur-
rounding regular islands is the main barrier to
diffusion of a trajectory outward. However, it
appears that the sticking time to a given sur-
face is the dominant process in impeding phase
space transport (Chirikov and Shepelyansky,
1984a,b; Murray, 1991). The departures of
diffusion from unimpeded random motion are
due to a small percentage of orbits that are
stuck around KAM surfaces bounding a region
of chaotic motion.
We may associate an escape past a particu-
lar cantorus with an "event" in an asteroid
orbit, since (going back to the two degree of
freedom analogy) successive cantori are rapidly
more porous, leading quickly to the global sea
of chaos and therefore wild excursions of the
orbit. We propose that this is a reasonable pos-
sibility for the mechanism displayed by the
outer-belt asteroids. The foregoing has been
well-established for simpler Hamiltonian sys-
tems (cf. Lichtenberg and Lieberman, 1992;
Wiggins, 1990).
An alternative view is that KAM tori are
"sticky." Karney (1983) was the first to
numerically study the stickiness of KAM tori,
finding that long-time correlation functions for
a version of the standard mapping are strongly
governed by the dynamics near KAM surfaces.
Chaotic Motion in the Outer Asteroid Belt
page 9

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